Towards a machine learning solution for hubble tension: Physics-Informed Neural Network (PINN) analysis of Tsallis Holographic Dark Energy in presence of neutrinos

Abstract We present a Physics-Informed Neural Network (PINN) framework for reconstructing the redshift-dependent Hubble parameter $$H(z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> within the Tsallis Holographic Dark Energy (THDE) model extended by massive neutrinos. In this approach, the modified Friedmann equation is incorporated into the neural network loss function, enabling training on Cosmic Chronometers data up to $$z \le 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The framework allows for the simultaneous estimation of the Hubble constant $$H_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , the neutrino density parameter $$\Omega _\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> </mml:math> , and the Tsallis non-extensivity index $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> . Uncertainty quantification is performed through dropout simulations, resulting in statistically consistent $$1\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> confidence bands. Our results show that the THDE+ $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> model, reconstructed via PINN, alleviates the statistical Hubble tension from the canonical $$\sim 5\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>5</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> level down to a range of $$0.5\sigma \le T \le 2.2\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.5</mml:mn> <mml:mi>σ</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>T</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2.2</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:math> , depending on the redshift sampling. Additionally, we constrain the total neutrino mass to $$\Sigma m_\nu &lt; 0.11\,\text {eV}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Σ</mml:mi> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:mn>0.11</mml:mn> <mml:mspace/> <mml:mtext>eV</mml:mtext> </mml:mrow> </mml:math> . A detailed comparison with the traditional Markov Chain Monte Carlo (MCMC) analysis demonstrates the consistency of both methods, while highlighting the competitiveness of the PINN-based THDE framework as a robust, data-driven approach for non-parametric cosmological inference within generalized thermodynamics.